<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 ;<br/>

<a NAME="E3:29"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a>, <a class="txt" href="fdiff_6.html#E1:29"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E4:29"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, <a class="txt" href="fdiff_6.html#E3:29"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="txt" href="fdiff_6.html#E2:29"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E7:29"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E8:29"/><i><font color="Green">E21</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E5:29"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fdiff_6.html#E2:29"><i><font color="Green">E17</font></i></a>, <a class="ref" href="fdiff_6.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E9:29"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<i><font color="Green">E24</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a>, <a class="txt" href="fdiff_6.html#E5:29"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fdiff_6.html#E1:29_1"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:29_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fdiff_6.html#E7:29"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D1">TAYLOR_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fdiff_6.html#E1:29_1"><i><font color="Green">E23</font></i></a></i>
;<br/>
<a NAME="E3:29_1"/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T59">SIN_COS:59</a></i>;<br/>
<a NAME="E4:29_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T49">PREPOWER:49</a></i>;<br/>
<b>hence </b><a NAME="E5:29_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="fdiff_6.html#E2:29_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fdiff_6.html#E2:29"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:29"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:29_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, <a class="txt" href="fdiff_6.html#E8:29"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fdiff_6.html#E2:29"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E3:29_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E9:29"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fdiff_6.html#E1:29_2"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:29_2"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T20">TAYLOR_1:20</a>, <a class="txt" href="fdiff_6.html#E2:29_2"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:29"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E3:29"><i><font color="Green">E18</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E12:29"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:29_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:29_3"/><i><font color="Green">E25</font></i>: 
 <a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T60">SIN_COS:60</a></i>;<br/>
<i><font color="Green">E26</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a>, <a class="txt" href="fdiff_6.html#E5:29"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fdiff_6.html#E1:29_3"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:29_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fdiff_6.html#E7:29"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D1">TAYLOR_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fdiff_6.html#E1:29_3"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="sin_cos.html#D27">SIN_COS:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="prepower.html#T54">PREPOWER:54</a>, <a class="txt" href="fdiff_6.html#E2:29_3"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prepower.html#K7">#Z</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
;<br/>
<a NAME="E4:29_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a>, <a class="txt" href="fdiff_6.html#E8:29"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fdiff_6.html#E2:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fdiff_6.html#E1:29_3"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E5:29_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_6.html#E9:29"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fdiff_6.html#E1:29_3"><i><font color="Green">E24</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E11:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fdiff_6.html#E1:29"><i><font color="Green">E16</font></i></a>, <a class="ref" href="fdiff_1.html#T28">FDIFF_1:28</a>, <a class="txt" href="fdiff_6.html#E1:29_3"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T20">TAYLOR_1:20</a>, <a class="txt" href="fdiff_6.html#E4:29_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="fdiff_6.html#E5:29_3"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="taylor_1.html#K1">#Z</a> 2</span>)</span> <a href="partfun1.html#K1">*</a> <a href="sin_cos.html#K27">exp_R</a> </span>)</span> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E1:29_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a>, <a class="txt" href="fdiff_6.html#E8:29"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fdiff_6.html#E2:29"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E5:29"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fdiff_6.html#E2:29"><i><font color="Green">E17</font></i></a>, <a class="ref" href="fdiff_6.html#T23" target="_self">Th23</a>, <a class="txt" href="fdiff_6.html#E1:29_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fdiff_6.html#E3:29_3"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:29_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T55">XCMPLX_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T55">SIN_COS:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:29_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_6.html#E2:29_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T4">TAYLOR_1:4</a></i>
;<br/>
<b>hence </b><a NAME="E7:29_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:29"/>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K2">log_</a> <a href="power.html#K8">number_e</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="fdiff_6.html#E11:29"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fdiff_6.html#E1:29"><i><font color="Green">E16</font></i></a>, <a class="ref" href="fdiff_1.html#T28">FDIFF_1:28</a></i>;<br/>


</div>
